\documentclass{letter}

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\begin{document}

a)

$$T_{\mu \nu} = ( \rho + p / c^2 ) u_\mu u_\nu - p \eta_{\mu \nu}$$

We see that the term $p u_\mu u_\nu / c^2 \sim \beta^2$, hence in the non-relativistic limit $\beta \rightarrow 0$ and we are left with

$$T_{\mu \nu} = \rho u_\mu u_\nu - p \eta_{\mu \nu}$$

now taking the derivative

$$f_\mu = \partial^\nu T_{\mu \nu} = \rho \partial^\nu (u_\mu u_\nu) - p \cancel{ \partial^\nu \eta_{\mu \nu} } - \eta_{\mu \nu} \partial^\nu p = p ( u_\mu \partial^\nu u_\nu + u_\nu \partial^\nu u_\mu ) - \partial_\mu p$$

now to find the derivatives, consider $u_\mu = \gamma \diff{}{t} x_\mu$, thus by interchanging the partial derivative $\partial^\nu$ we obtain $$\partial^\nu u_\nu = \gamma \diff{}{t} \partial^\nu x_\nu = \gamma \diff{}{t} \left( \frac{1}{c} \pdiff{}{t} (ct) - \nabla \cdot \vec x \right) = \gamma \diff{}{t} \left( 1 - \nabla \cdot \vec x \right) = - \gamma \nabla \cdot \vec u$$ now $\partial^\nu u_\mu$ is clearly some sort of matrix, and we are motivated by the observation that, in 3 spatial dimensions, $\nabla_i r_j = \delta_{ij}$ $$\partial^\nu u_\mu $$

b)

Again in nonrelativistic limit, we have $$T_{\mu \nu}  = \rho u_\mu u_\nu - \eta_{\mu \nu} p.$$ Set $\mu =  i$ and sum over $\nu$:

\begin{eqnarray*}
f_i &=& \partial^\nu T_{i \nu}  = \partial^\nu \left( \rho u_i u_\nu \right) - \partial_i p \\
&=& \partial^0 \left( \rho u_i u_0 \right) + \nabla_j \left( \rho u_i u_j \right) + \nabla_i p \\
&=& \frac{1}{c} \pdiff{}{t} \left( \rho u_i c \right) + \rho ( \vec u \cdot \nabla ) u_i  + u_i \nabla \cdot ( \rho \vec u ) + \nabla_i p \\
&=& u_i \pdiff{}{t} \rho + \rho \pdiff{}{t} u_i + \rho ( \vec u \cdot \nabla ) u_i + u_i \nabla \cdot ( \rho \vec u ) + \nabla_i p \\
&=& \rho \pdiff{}{t} u_i - \rho ( \vec u \cdot \nabla ) u_i + \nabla_i p \\
\Rightarrow \rho \left( \pdiff{}{t} + \rho (\vec u \cdot \nabla) \right) \vec u &=& - \nabla p +  \vec f
\end{eqnarray*}

where we have used the continuity equation to replace $\pdiff{}{t} \rho = - \nabla \cdot (\rho \vec u)$ thus cancelling the term $u_i \nabla \cdot ( \rho \vec u )$.

\end{document}